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# DS questions about standard deviation

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29 Jun 2011, 09:38
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Bunuel wrote:
Lately, many questions were asked about the standard deviation. So I'm posting here my collection of DS problems on SD, plus some tips about it.

A. I was assured MANY TIMS, by various GMAT tutors, that GMAT won't ask you to actually calculate SD, but rather to understand the concept of it. Though KNOWING how it's calculated helps in understanding the concept.
B. During the real GMAT it's highly unlikely to get more than one ot two question on SD (as on combinatorics), actually you may see none, so do not spend too much of your preparation time on it, it's better to concentrate on issues you'll definitely face on G-day.

Many questions below are easy, some are tough, but anyway they are good to master in solving SD problems. I'll post OA after some discussions. Please provide your way of thinking along with the answer. Thanks.

Here we go:

1. What is the standard deviation of Company R’s earnings per month for this year?
(1) The standard deviation of Company R’s earnings per month in the first half of this year was $2.3 million. (2) The standard deviation of Company R’s earnings per month in the second half of this year was$3.9 million.

2. What is the standard deviation of Q, a set of consecutive integers?
(1) Q has 21 members.
(2) The median value of set Q is 20.

3. Lifetime of all the batteries produced by certain companies have a distribution which is symmetric about mean m. If the distribution has a standard deviation of d , what percentage of distribution is greater than m+d?
(1) 68 % of the distribution in the interval from m-d to m+d, inclusive
(2) 16% of the distribution is less than m-d

4. Question deleted

5. List S and list T each contain 5 positive integers, and for each list the average of the integers in the list is 40. If the integers 30,40 and 50 are in both lists , is the standard deviation of the integers in list S greater than the standard deviation of the integers in list T?
(1)The integer 25 is in list S
(2)The integer 45 is in list T

6. Set T consists of odd integers divisible by 5. Is standard deviation of T positive?
(1) All members of T are positive
(2) T consists of only one member

7. Set X consists of 8 integers. Is the standard deviation of set X equal to zero?
(1) The range of set X is equal to 3
(2) The mean of set X is equal to 5

8. {x,y,z}
If the first term in the data set above is 3, what is the third term?

(1) The range of this data set is 0.
(2) The standard deviation of this data set is 0.

9. Question deleted

10. A scientist recorded the number of eggs in each of 10 birds' nests. What was the standard deviation of the numbers of eggs in the 10 nests?
(1) The average (arithmetic mean) number of eggs for the 10 nests was 4.
(2) Each of the 10 nests contained the same number of eggs.

11. During an experiment, some water was removed from each of the 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?
(1) For each tank, 30% of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.
(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.

CALCULATING STANDARD DEVIATION OF A SET {x1, x2, ... xn}:
1. Find the mean, m, of the values.
2. For each value xi calculate its deviation (xi-m) from the mean.
3. Calculate the squares of these deviations.
4. Find the mean of the squared deviations. This quantity is the variance.
5. Take the square root of the variance. The quantity is th SD.

TIPS:
1. |Median-Mean| <= SD.

2. Variance is the square of the standard deviation.

3. If Range or SD of a list is 0, then the list will contain all identical elements. And vise versa: if a list contains all identical elements then the range and SD of a list is 0. If the list contains 1 element: Range is zero and SD is zero.

4. SD is always >=0. SD is 0 only when the list contains all identical elements (or which is same only 1 element).

5. Symmetric about the mean means that the shape of the distribution on the right and left side of the curve are mirror-images of each other.

6. If we add or subtract a constant to each term in a set:
Mean will increase or decrease by the same constant.
SD will not change.

7. If we increase or decrease each term in a set by the same percent:
Mean will increase or decrease by the same percent.
SD will increase or decrease by the same percent.

8. Changing the signs of the element of a set (multiplying by -1) has no effect on SD.

9. The SD of any list is not dependent on the average, but on the deviation of the numbers from the average. So just by knowing that two lists having different averages doesn't say anything about their standard deviation - different averages can have the same SD.

You can also check collection of PS questions of SD at: ps-questions-about-standard-deviation-85897.html

Hi Bunuel,

I find the collection of questions that you posted on Standard Deviation very helpful. Do you post questions like this on all the other topic as well?

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01 Jul 2011, 16:00
Great collection of excellent questions...

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18 Mar 2012, 11:43
[quote="Bullet"]I'll try to provide some explanation of the questions with reference to answers, please correct me if i'm wrong.

Thanks

1. What is the standard deviation of Company R’s earnings per month for this year?
(1) The standard deviation of Company R’s earnings per month in the first half of this year was $2.3 million. (2) The standard deviation of Company R’s earnings per month in the second half of this year was$3.9 million.

Reason : As none of the information above is adequate to find the SD, therefore SD is not possible to calculate

Wanted to make a point here, please clarify if im wrong that SD is calculated as follows:
standard \ deviation= \sqrt{variance} = \sqrt{\frac{\sum(m-x_i)^2}{N}}.

and the two statements give me the values inside the function hence can be used to calculate the SD for the entire year...

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08 Jul 2012, 00:49
1. What is the standard deviation of Company R’s earnings per month for this year?
(1) The standard deviation of Company R’s earnings per month in the first half of this year was $2.3 million. (2) The standard deviation of Company R’s earnings per month in the second half of this year was$3.9 million.

11. During an experiment, some water was removed from each of the 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?
(1) For each tank, 30% of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.
(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.

Hello Bunnel,

Please explain Q 1 If we combine the two statements and then take the weighted average of the two Standard deviations, wouldn't that give us the STD Dev for the whole year?

Please explain Stat (2) for Q 11, How do we prove it insuff?

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08 Jul 2012, 03:50
teal wrote:
1. What is the standard deviation of Company R’s earnings per month for this year?
(1) The standard deviation of Company R’s earnings per month in the first half of this year was $2.3 million. (2) The standard deviation of Company R’s earnings per month in the second half of this year was$3.9 million.

11. During an experiment, some water was removed from each of the 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?
(1) For each tank, 30% of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.
(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.

Hello Bunnel,

Please explain Q 1 If we combine the two statements and then take the weighted average of the two Standard deviations, wouldn't that give us the STD Dev for the whole year?

Please explain Stat (2) for Q 11, How do we prove it insuff?

11. During an experiment, some water was removed from each of the 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?

(1) For each tank, 30% of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.
(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.

You should know that:
If we add or subtract a constant to each term in a set:
Mean will increase or decrease by the same constant.
SD will not change.

If we increase or decrease each term in a set by the same percent (multiply all terms by the constant):
Mean will increase or decrease by the same percent.
SD will increase or decrease by the same percent.

You can check it yourself:
SD of a set: {1,1,4} will be the same as that of {5,5,8} as second set is obtained by adding 4 to each term of the first set.

That's because Standard Deviation shows how much variation there is from the mean. And when adding or subtracting a constant to each term we are shifting the mean of the set by this constant (mean will increase or decrease by the same constant) but the variation from the mean remains the same as all terms are also shifted by the same constant.

So according to this rules statement (1) is sufficient to get new SD, it'll be 30% less than the old SD so 7. As for statement (2) it's clearly insufficient as knowing mean gives us no help in getting new SD.

As for 1: we only know that the standard deviation of some set with 6 data points is 2.3 and the standard deviation of other set with 6 data points is 3.9. From that info we cannot calculate the standard deviation of the combined set (you cannot just take an average of these two values).
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08 Jul 2012, 20:12
Thanks Bunnel, please explain q 1 as well as listed in my post above.

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27 Sep 2013, 05:44
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05 Oct 2013, 08:43
8. {x,y,z}
If the first term in the data set above is 3, what is the third term?
(1) The range of this data set is 0.
(2) The standard deviation of this data set is 0.

i got B, maybe you can help me out with this

for stmt 1 -> {x,y,z} is a set and it does not mention of the order, so it can be {3,1,3} or {3,3,1} and still have the range of 0. Therefore stmt 1 would be insufficient.

Stmt 2 is sufficient cause all the values have to be the same.

Maybe i'm missing something but as per OA the assumption is {x,y,z} is in ascending order.

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05 Oct 2013, 11:22
Whatup780 wrote:
8. {x,y,z}
If the first term in the data set above is 3, what is the third term?
(1) The range of this data set is 0.
(2) The standard deviation of this data set is 0.

i got B, maybe you can help me out with this

for stmt 1 -> {x,y,z} is a set and it does not mention of the order, so it can be {3,1,3} or {3,3,1} and still have the range of 0. Therefore stmt 1 would be insufficient.

Stmt 2 is sufficient cause all the values have to be the same.

Maybe i'm missing something but as per OA the assumption is {x,y,z} is in ascending order.

The range = largest - smallest.
The range of both {3,1,3} and {3,3,1} is 2, not 0.

The range is 0 means that the elements of a set are all equal.
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26 Nov 2013, 09:10
Bunuel,
In this question
2. What is the standard deviation of Q, a set of consecutive integers?
(1) Q has 21 members.
(2) The median value of set Q is 20.

We know nothing about statement 1.As in we have 21 members yes but we don't know their values,how can we then find their SD?
Isn't that the reason we disqualified statement 2?

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26 Nov 2013, 09:15
mumbijoh wrote:
Bunuel,
In this question
2. What is the standard deviation of Q, a set of consecutive integers?
(1) Q has 21 members.
(2) The median value of set Q is 20.

We know nothing about statement 1.As in we have 21 members yes but we don't know their values,how can we then find their SD?
Isn't that the reason we disqualified statement 2?

Two very important properties of standard deviation:

If we add or subtract a constant to each term in a set:
Mean will increase or decrease by the same constant.
SD will not change.

If we increase or decrease each term in a set by the same percent (multiply all terms by the constant):
Mean will increase or decrease by the same percent.
SD will increase or decrease by the same percent.

You can try it yourself:
SD of a set: {1,1,4} will be the same as that of {5,5,8} as second set is obtained by adding 4 to each term of the first set.

That's because Standard Deviation shows how much variation there is from the mean. And when adding or subtracting a constant to each term we are shifting the mean of the set by this constant (mean will increase or decrease by the same constant) but the variation from the mean remains the same as all terms are also shifted by the same constant.

Back to the original question:

If Q is a set of consecutive integers, what is the standard deviation of Q?

(1) Set Q contains 21 terms --> SD of ALL sets with 21 consecutive integers will be the same, as any set of 21 consecutive integers can be obtained by adding constant to another set of 21 consecutive integers. For example: set of 21 consecutive integers {4, 5, 6, ..., 24} can be obtained by adding 4 to each term of another set of 21 consecutive integers: {0, 1, 2, ..., 20}. So we can calculate SD of {0, 1, 2, ..., 20} and we'll know that no matter what our set actually is, its SD will be the same. Sufficient.

(2) The median of set Q is 20. We don't know how many terms are in Q. Not sufficient.

Hope it's clear.
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26 Nov 2013, 10:05
Bunuel,
Thanks i understand that now

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16 Dec 2014, 07:53
Question about #1. Maybe I've missed a fundamental of standard deviation.

1. What is the standard deviation of Company R’s earnings per month for this year?
(1) The standard deviation of Company R’s earnings per month in the first half of this year was $2.3 million. (2) The standard deviation of Company R’s earnings per month in the second half of this year was$3.9 million.

Based on statement 1, we can say the variance is 2.3^2. And this variance is just the squares of the difference between each value and the mean divided by the number of data points. In this case it would be the sum of the differences in each month's earnings divided by the mean earning divided by 6. Can't we break down each statement to find the sum of the squares of the differences for each six months and divide by 12?

For example, say we have the integers 1-12, representing the earnings from Jan to Dec. So, we have 1-6 for the first half and 7-12 for the second half. We could calculate the squares of the differences for each half, sum them up and divide the whole number by 12. Is this incorrect? Can someone clarify?

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16 Dec 2014, 08:59
intheend14 wrote:
Question about #1. Maybe I've missed a fundamental of standard deviation.

1. What is the standard deviation of Company R’s earnings per month for this year?
(1) The standard deviation of Company R’s earnings per month in the first half of this year was $2.3 million. (2) The standard deviation of Company R’s earnings per month in the second half of this year was$3.9 million.

Based on statement 1, we can say the variance is 2.3^2. And this variance is just the squares of the difference between each value and the mean divided by the number of data points. In this case it would be the sum of the differences in each month's earnings divided by the mean earning divided by 6. Can't we break down each statement to find the sum of the squares of the differences for each six months and divide by 12?

For example, say we have the integers 1-12, representing the earnings from Jan to Dec. So, we have 1-6 for the first half and 7-12 for the second half. We could calculate the squares of the differences for each half, sum them up and divide the whole number by 12. Is this incorrect? Can someone clarify?

Notice that the means of the first half and the second half are not necessarily equal, so this won't work.
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16 Aug 2015, 21:09
This is a wonderful post with most of the things one should know about SD!

(2) The standard deviation of Company R’s earnings per month in the second half of this year was $3.9 million. Answer: E. Reason : As none of the information above is adequate to find the SD, therefore SD is not possible to calculate 2. What is the standard deviation of Q, a set of consecutive integers? (1) Q has 21 members. (2) The median value of set Q is 20. Answer: A. Reason: A is the right answer as we know from the Stmt 1 the total number of members in the set and consecutive integers shows that they're not identical so we can calculate the SD from this statement. Stmt 2 is not sufficient as with median we can't calculate SD. 3. Lifetime of all the batteries produced by certain companies have a distribution which is symmetric about mean m. If the distribution has a standard deviation of d , what percentage of distribution is greater than m+d? (1) 68 % of the distribution in the interval from m-d to m+d, inclusive (2) 16% of the distribution is less than m-d Answer: D. Reason: This is the same problem which is also asked in the PS problem. If the distribution is symmetric about the mean then it means that 50% of the population is above and below the mean. Stmt 1: if 68% population is within one SD of the mean then 34% is either above or below the mean. Out of 100, 32% population is outside 1SD of the mean so 16% of the population is m+d as well as 16% as m-d. Stmt2: also reflects the same thing that 16% population is below m-d so both of them are sufficient. 4. Set A and B are 2 sets of numbers. A has a standard deviation 3 and a mean 5. B has the same mean but standard deviation 4. Can we find the standard deviation of the set A U B ? If no, then is it possible with the following additional information? (1) Both sets have the same number of members each (let’s say 4) (2) Both sets have distinct members (no number is common to both sets). Answer: A. Reason: A is the option as with the help of the Stmt 1 we know the total members of the set however Stmt 2 means distinct members so the set wont be AUB then it will be AnB. 5. List S and list T each contain 5 positive integers, and for each list the average of the integers in the list is 40. If the integers 30,40 and 50 are in both lists , is the standard deviation of the integers in list S greater than the standard deviation of the integers in list T? (1)The integer 25 is in list S (2)The integer 45 is in list T Answer: C. Reason: Stmt 1 S = {25,30,40,50,x} Mean = 40 then x = 55 Stmt 2 T = {x,30,40,45,50} Mean = 40 then x= 35 So with the help of both stmts Set S has greater SD than T, that is why C is the answer 6. Set T consists of odd integers divisible by 5. Is standard deviation of T positive? (1) All members of T are positive (2) T consists of only one member Answer: B. Stmt 1 is not sufficient as all members are +ive but they may be identical so SD will be zero. Stmt 2 has only one member so SD is zero so that is why B is the answer 7. Set X consists of 8 integers. Is the standard deviation of set X equal to zero? (1) The range of set X is equal to 3 (2) The mean of set X is equal to 5 Answer: A. Stmt 1: if the range of the set is not equal to zero then SD will not be zero it will be greater than zero. Stmt 2: Mean can be 5 but set may contain identical elements. So A is right but SD will not be zero. 8. {x,y,z} If the first term in the data set above is 3, what is the third term? (1) The range of this data set is 0. (2) The standard deviation of this data set is 0. Answer: D. Both stmt are sufficient as with Stmt 1 we know that range is zero so the third term will be 3 and with the stmt 2 it shows that SD = 0 so third term will be 3. 9. Question deleted 10. A scientist recorded the number of eggs in each of 10 birds' nests. What was the standard deviation of the numbers of eggs in the 10 nests? (1) The average (arithmetic mean) number of eggs for the 10 nests was 4. (2) Each of the 10 nests contained the same number of eggs. Answer: B. May be i don't understand the question properly, B is the right choice as each of the nest have identical eggs therefore SD = 0 or i missed something 11. During an experiment, some water was removed from each of the 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment? (1) For each tank, 30% of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment. (2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons. Answer: A. As 30% of the volume is removed from all the tanks so SD remains same. therefore stmt 1 is sufficient. Does removing 30% brings the SD down by 30 % of the original or it will remain the same? Experts, pls clarify... Kudos [?]: 8 [0], given: 219 Math Expert Joined: 02 Sep 2009 Posts: 42356 Kudos [?]: 133196 [0], given: 12439 Re: DS questions about standard deviation [#permalink] ### Show Tags 29 Dec 2015, 01:27 ArunpriyanJ wrote: Bullet wrote: I'll try to provide some explanation of the questions with reference to answers, please correct me if i'm wrong. Thanks ANSWERS (OA): 1. What is the standard deviation of Company R’s earnings per month for this year? (1) The standard deviation of Company R’s earnings per month in the first half of this year was$2.3 million.
(2) The standard deviation of Company R’s earnings per month in the second half of this year was \$3.9 million.

Reason : As none of the information above is adequate to find the SD, therefore SD is not possible to calculate

2. What is the standard deviation of Q, a set of consecutive integers?
(1) Q has 21 members.
(2) The median value of set Q is 20.

Reason: A is the right answer as we know from the Stmt 1 the total number of members in the set and consecutive integers shows that they're not identical so we can calculate the SD from this statement. Stmt 2 is not sufficient as with median we can't calculate SD.

3. Lifetime of all the batteries produced by certain companies have a distribution which is symmetric about mean m. If the distribution has a standard deviation of d , what percentage of distribution is greater than m+d?
(1) 68 % of the distribution in the interval from m-d to m+d, inclusive
(2) 16% of the distribution is less than m-d

Reason: This is the same problem which is also asked in the PS problem. If the distribution is symmetric about the mean then it means that 50% of the population is above and below the mean.

Stmt 1: if 68% population is within one SD of the mean then 34% is either above or below the mean. Out of 100, 32% population is outside 1SD of the mean so 16% of the population is m+d as well as 16% as m-d.

Stmt2: also reflects the same thing that 16% population is below m-d so both of them are sufficient.

4. Set A and B are 2 sets of numbers. A has a standard deviation 3 and a mean 5. B has the same mean but standard deviation 4. Can we find the standard deviation of the set A U B ? If no, then is it possible with the following additional information?
(1) Both sets have the same number of members each (let’s say 4)
(2) Both sets have distinct members (no number is common to both sets).

Reason: A is the option as with the help of the Stmt 1 we know the total members of the set however Stmt 2 means distinct members so the set wont be AUB then it will be AnB.

5. List S and list T each contain 5 positive integers, and for each list the average of the integers in the list is 40. If the integers 30,40 and 50 are in both lists , is the standard deviation of the integers in list S greater than the standard deviation of the integers in list T?
(1)The integer 25 is in list S
(2)The integer 45 is in list T

Reason:
Stmt 1 S = {25,30,40,50,x}
Mean = 40
then x = 55

Stmt 2 T = {x,30,40,45,50}
Mean = 40
then x= 35

So with the help of both stmts Set S has greater SD than T, that is why C is the answer

6. Set T consists of odd integers divisible by 5. Is standard deviation of T positive?
(1) All members of T are positive
(2) T consists of only one member

Stmt 1 is not sufficient as all members are +ive but they may be identical so SD will be zero.
Stmt 2 has only one member so SD is zero

so that is why B is the answer

7. Set X consists of 8 integers. Is the standard deviation of set X equal to zero?
(1) The range of set X is equal to 3
(2) The mean of set X is equal to 5

Stmt 1: if the range of the set is not equal to zero then SD will not be zero it will be greater than zero.
Stmt 2: Mean can be 5 but set may contain identical elements.

So A is right but SD will not be zero.

8. {x,y,z}
If the first term in the data set above is 3, what is the third term?
(1) The range of this data set is 0.
(2) The standard deviation of this data set is 0.

Both stmt are sufficient as with Stmt 1 we know that range is zero so the third term will be 3 and with the stmt 2 it shows that SD = 0 so third term will be 3.

9. Question deleted

10. A scientist recorded the number of eggs in each of 10 birds' nests. What was the standard deviation of the numbers of eggs in the 10 nests?
(1) The average (arithmetic mean) number of eggs for the 10 nests was 4.
(2) Each of the 10 nests contained the same number of eggs.

May be i don't understand the question properly, B is the right choice as each of the nest have identical eggs therefore SD = 0 or i missed something

11. During an experiment, some water was removed from each of the 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?
(1) For each tank, 30% of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.
(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.

As 30% of the volume is removed from all the tanks so SD remains same. therefore stmt 1 is sufficient.

Does removing 30% brings the SD down by 30 % of the original or it will remain the same?

Experts, pls clarify...

If we add or subtract a constant to each term in a set:
Mean will increase or decrease by the same constant.
SD will not change.

If we increase or decrease each term in a set by the same percent (multiply all terms by the constant):
Mean will increase or decrease by the same percent.
SD will increase or decrease by the same percent.

So according to this rules statement (1) is sufficient to get new SD, it'll be 30% less than the old SD so 7.

Check the discussion of this question HERE.
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24 May 2016, 01:53
Hi Bunuel,

I find the collection of questions that you posted on Standard Deviation very helpful. Do you post questions like this on all the other topic as well?

Thanks,Ruben.

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